Murray riding mower with a briggs and stratton 16 hp engine will not start after I washed the mower off . What should I check? Only a click sound when I turn the key.

I am guessing that some of the water got into the engine when you washed it off. It probably has water damage.

Check the battery, and you can use a piece of battery cable sized wire to jump across both of the top (large) post on the solenoid to see if it starts then. If jumping across the solenoid makes the engine turn over, then you need a solenoid.